METHODS   OP    LOCATION 

FOR 

RAILWAY  ENGINEERS, 

BY 

S.    W.    MIFFLIN, 

CIVIL    ENGINEER. 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

GIFT    OF 


Class 


I 


-       ' 


• 


. 


METHODS   OF  L 


DESCIBING  AND  ADJUSTI 


RAILWAY  CURVES  ANT)  TANGENTS, 


AS  PRACTISED  BY  THE 


ENGINEERS  OF  PENNSYLVANIA, 

REVISED  AND  EXTENDED 

BY  SAMUEL  W.  MIFFLIN, 
CIVIL  ENGINEER. 


PHILADELPHIA: 
EDWARD    C.   BIDDLE, 

23  Minor  Street, 

1837. 


Entered  according  to  the  act  of  congress,  in  the  year  1837,  by  EDWARD'  C. 
BIDDLE  in  the  clerk's  office  of  the  district  court  of  the  eastern  district  of 
Pennsylvania. 


Philadelphia: 

T.  K.  &  P.  G.  Collins,  Printers, 
No.  1  Lodge  Alley. 


PREFACE. 


IN  submitting  this  work  to  the  public,  I  do 
not  wish  to  claim  for  myself  its  exclusive 
authorship.  A  few  of  the  first  solutions  were 
the  work  of  my  comrades  on  the  Pennsylvania 
Railway;  these  falling  into  my  hands  suggested 
the  idea  of  a  complete  series  of  Geometric 
solutions  applicable  to  all  cases  that  might 
occur. 

In"  such  a  work  I  considered  it  of  import- 
ance to  dispense  with  all  difficult  calculations, 
and  even  with  tabular  statements,  which  can- 
not be  committed  to  memory..  In  this  I  have 
happily  succeeded ;  there  is  nothing  in  the  fol- 
lowing pages  which  may  not  be  remembered  by 
an  assistant  after  a  short  practice,  and  executed 
in  the  field  even  if  the  book  be  left  at  home. 

There  are  a  few  instances,  however,  in  which 
a  table  of  chords  may  facilitate  operations,'and 
one  has  therefore  been  placed  upon  the  last 
page  for  the  use  of  those  who  prefer  it. 


202371 


V 


• 


I 


EXPLANATIONS. 


1.  SINCE  all  the  curves  described  in  this  work  are  cir- 
cular, the  words  curve  and  circle  will  be  used  indis- 
criminately. 

2.  All    measurements  in    this    work    are  referred    to 
some  chord  of  convenient  length  as  a  unit,  which, 
may  be  either  the  common  four  pole  chain  of  100 
links,  or  one  of  100  feet,  and  for  brevity  sake  the 
word  chain  will  be  used  to  designate  such  chord. 

3.  The    angle  subtended  by  the  above    chord  at  the 
centre  of  the  circle  is  called  the  degree  of  curvature, 
or  simply  the  curvature. 

4.  The  letters  m  and  n  are  used  to  express  degrees  of 
curvature,  and  when  both  are  used  m  is  the  greatest, 
that  is,  it  belongs  to  the  smallest  circle. 

5.  A  central  angle  is  that  which  a  chord  subtends  at 
the  centre  of  the  circle. 

6.  A  circumferential  angle  is  that  whiQh  a  chord  sub- 
tends at  any  point  in  the  circumference. 

7.  A  tangential  angle  is  the  smallest  angle  made  by  a 
chord  at  its  extremity,  with  a  tangent  to  the  curve 
at  that  extremity. 

8.  A  compound  curve  is  composed  of  two  curves  of 
different  radii  turning  in  the  same  direction,  having 
a  common  tangent  at  their  point  of  meeting. 

9.  This  point  of  meeting  is  called  the  point  of  com- 
pound curvature,  or  simply  P.  C.  C. 

10.  A  reversed  curve  is  composed  of  two  curves  turn- 
ing in  opposite  directions  and  having  a  common 
tangent  at  their  point  of  meeting. 

1* 


6          .  EXPLANATIONS. 

I 

11.  This  point  is  called  P.  R.  C.,  or  point  of  reversed 
curvature. 

12.  JL  differential  curve  is  one  whose  radius  is  equal 
to  the  difference  between  the  radii  of  any  two  curves 
to  which  it  is  applied. 

13.  An  integral  curve  is  one  whose  radius  equals  the 
sum  of  the  radii  of  two  other  curves. 

14.  Equivalent  arcs  or  curves  are  such  as  subtend  equal 
central  angles. 

15.  Corresponding  points  in  different  circles  are  any 
points,  where  the  tangents  and  of  course  the  radii 
are  parallel. 

16.  The  terms  origin  and  termination  are  used  in  re- 
ference to  the  course  of  location.     The  termination 
of  a  tangent  being  the  point  where  a  curve  is  com- 
menced, and  the  origin  of  the  next  tangent  the  point 
where  the  curve  terminates. 

17.  The  origin  is  also  called  the  point  of  curve,  or 
point  of  tangent,  or  simply  P.  C.  or  P.  T. 


PRELIMINARY  PROPOSITIONS. 


Condensed  from  Euclid,  Book  Third. 

1.  THE  Angle  AFB  (Fig.  1st,)  subtended  by  any 
chord  AB  at  the  centre,  is  double  the  angle  AEB  at 
any  part  of  the  circumference  on  the  same  side  of  the 
chord. 

•  2.  Equal  chords  AB,  BC,  CD,  subtend  equal  angles 
whether  at  the  centre  or  circumference. 

3.  The  angle  BAG  formed  by  any  chord  AB  with 
a  tangent  at  either  extremity,  is  equal  to  half  the  angle 
AFB  at  the  centre,  or  to  the  angle  AEB  at  the  cir- 
cumference. 

4.  The  exterior  angle  HBC  formed  by  two  equal 
chords  AB,  BC,  is  equal  to  the  central  angle  AFB,  or 
CFB,  or  double  the  tangential  angle  GAB. 

5.  The  exterior  angle  LAI  of  two  unequal  chords, 
LA,   AB,   is  equal  to  half  the  sum  of  their  central 
angles  LFB,  or  to  the  sum  of  their  tangential  angles 
LAM  +  MAI  or  GAB. 

6.  The  exterior  angle  of  any  two  chords  AN,  NB, 
is  equal  to  one  half  the  central  angle  of  AB,  or  its 
exterior  angle  with  its  equal  BC. 

7.  The   exterior  angle  KOC,  of  any  two  tangents 
CO,  BK,   is   equal  to  the  central  angle  BFC    of  the 
chord  BC,  which  joins  their  point  of  contact 


METHODS  OF  LOCATION 


The  following  Propositions,  which  are  likewise  used 
in  this  work,  although  not  strictly  correct,  are 
sufficiently  so  for  all  purposes  of  Location. 

8.  The  central,  circumferential,  and  tangential  angles 
of  chords  of  unequal  lengths  are  directly  as  the  lengths. 

9.  The  radii  of  circles  are  directly  as  their  degrees 
of  curvature. 

10.  The  radius  of  a  circle  is  half  the  circumference 
divided  by  3.1416. 

11.  If  the  chord  of  one  degree  be  taken  as  a  unit, 
the  circumference  may  be  considered  equal  to  360. 

180 
Hence,  the  radius  is  equal  to  Q  =  57. 30  and  by 

O«  JL  ~t  A  D 

57.3 
proposition  9  the  radius  of  any  other  circle  is 

NOTE. — The  8th,  9th  and  llth  propositions  are  true  to  the  second 
place  of  decimals,  so  long  as  ra  is  not  greater  than  10°,  which  is 
double  what  is  required  in  ordinary  cases, 


THE 
I   UNIVER- 

Of 

»£SUE§£.* 


E 

F 

-D 


FOR  RAILWAY  ENGINEERS.  9 

PRELIMINARY  EXERCISES. 

In  the.  Use  of  the  Transit. 

THE  Transit  is  an  instrument  invented  and  manu- 
factured by  W  J.  Young  of  Philadelphia.  It  is  in 
many  respects  more  convenient  than  the  Goniometer  or 
Theodolite,  and  being  the  instrument  to  which  I  have 
been  most  accustomed,  I  have  adapted  the  phraseology 
of  this  treatise  to  its  use.  There  is,  however,  no  diffi- 
culty in  solving  all  the  propositions  in  this  series  with 
either  of  the  other  instruments  above  mentioned. 

It  is  not  my  purpose  to  give  a  description  of  the 
Transit  instrument,  but,  supposing  the  student  to  have 
one  before  him,  and  to  be  acquainted  with  the  uses  of 
its  various  parts,  I  shall  proceed  to  describe'  some  of 
its  most  common  applications. 

PROPOSITION  I. 

To  adjust  the  vertical  hair  of  the  Telescope  so  that 
the  Lines  of  sight  forward  and  backwards  shall  be 
parts  of  the  same  Straight  Line. 

CHOOSE  a  piece  of  perfectly  level  ground,  from  500 
to  800  feet  long,  and  clear  of  all  obstruction  to  the 
sight,  set  the  instrument  in  the  middle  as  at  A,  Fig. 
2d,  level  and  clamp  it,  and  with  the  tangent  screws 
bring  the  sight  to  bear  upon  a  chain-pin  or  any  other 
suitable  object  which  an  assistant  must  hold  at  B. 

Then  reverse  the  Telescope  on  its  axis  and  set  up 
another  pin  in  the  opposite  direction,  and  at  the  same 
distance  as  B  is  from  A:  if  the  instrument  be  out  of 
adjustment  this  will  not  fall  in  the  line  AB  produced 
but  on  one  side  of  it  as  at  C. 

Now  unscrew  the  clamp  and  without  touching  the 
Telescope  reverse  the  Transit  on  its  axis,  and  fix  the 
sight  upon  B  as  before,  and  screw  up  the  clamp. 
Again  reverse  the  Telescope  and  set  up  a  third  pin, 
which  will  n.ow  fall  upon  the  point  D  precisely,  as  far 


10  METHODS  OF  LOCATION 

to  the  right  of  AE,  as  C  is  to  the  left.  Divide  accu- 
rately the  distance  between  C  and  D,  and  set  up  a 
fourth  pin  at  E  ;  B,  A,  and  E  will  then  be  in  the  same 
straight  line.  Now  remove  the  pin  from  C,  and  set  it 
up  at  F,  precisely  in  the  middle  of  E  D,  and  with  the 
ad  justing  pin  remove  the  vertical  hair  until  it  coincides 
with  F,  then  with  the  tangent  screws  fix  the  sight 
upon  E  and  reverse  the  Telescope.  If  the  operation 
has  been  carefully  performed,  the  sight  will  strike  the 
chain-pin  at  B  and  the  adjustment  is  effected. 

It  generally  happens,  however,  that  a  second  slight 
movement  of  the  hair  is  necessary  to  perfect  the  ope- 
ration, which  should  be  tested  by  several  reversions  on 
the  axis  of  both  Transit  and  Telescope,  until  the  coin- 
cidence of  the  hair  with  B  and  E  is  fully  established, 

PROPOSITION  II. 

To  discover  whether  the   Telescope  revolves  truly  in 
the  Meridian. 

AFTER  completing  the  adjustment  by  the  last  pro- 
position, choose  a  steeple  or  any  other  lofty  object 
upon  whose  top  a  steady  and  accurate  sight  can  be 
obtained  ;  set  the  instrument  as  near  to  its  base  as 
possible,  and  after  leveling  and  clamping  it,  fix  the 
sight  upon  the  top  of  the  object  and  turn  the  head  of 
the  Telescope  in  the  opposite  direction,  so  as  to  bear 
upon  the  ground  at  some  convenient  distance  from  the 
instrument,  and  set  up  a  pin. 

Then  reverse  the  Transit  on  its  axis  and  take  sight 
as  before  to  the  top  of  the  steeple  :  again  turn  the  head 
of  the  Telescope  towards  the  pin  just  set  up,  and  if 
the  vertical  hair  coincides  with  it,  the  instrument  is 
sound,  but  if  not,  half  the  distance  between  them  will 
be  the  error.  As  this  inaccuracy  is  always  the  result  of 
accident,  a  blow,  or  a  fall,  there  is  no  method  of  remov- 
ing it  in  the  field;  when  discovered  it  should  be  sent 
to  the  maker  for  repairs. 


D 


Fig;.  4 


A  B 


A  B 


» 


FOR  RAILWAY  ENGINEERS.  11 


PROPOSITION  III. 


To  Measure  the  Jingle  between  any  1wo  Lines  or 
Deflection  from  one  Line  to  another. 

IN  performing  this  operation,  it  is  necessary  to  bear 
in  mind  the  course  of  survey  :  thus,  if  the  survey  pro- 
ceeds from  A  to  B,  Fig.  3d,  and  afterwards  from  B  to 

C,  then  the  angle  required  will  be  CBD,  but   if  the 
course  of  survey  is  from  D  to  B,  and  afterwards  from 
B  to  C,  then  the  angle  required  is  ABC.     To  measure 
the  angle  proceed  as  follows  :  place  the  instrument 
over  B,  and  set  the  index  to  Zero,  then  take  a  back 
sight  from  B  to  A,  and  reverse  the   Telescope:  turn 
the  rack  until  the  sight  bears  upon  C,  and  the  index 
will  then  show  the  angle  DBC  required. 

CASE  2d. —  When  the  intersection  of  two  lines  AB 
and  CD,  Fig.  4th,  are  inaccessible.  Place  the  instru- 
ment over  B,  set  the  index  to  Zero,  and  take  a  back 
sight  to  A,  then  reverse  the  Telescope,  turn  the  rack  and 
take  sight  upon  C  ;  screw  the  rack  fast,  and  the  instru- 
ment to  C  ;  take  a  back  sight  from  C  to  B  :  again  reverse 
the  Telescope  and  turn  the  rack  till  the  sight  falls  upon 

D.  The  index  will  then  show  the  required  angle. 
CASE  3d. —  When  the  ground  is  so   encumbered 

that  no  part  of  CD  can  be  seen  from  AB.  Take  a 
point  E,  Fig.  5th,  from  which  both  lines  may  be  seen, 
and  deflect  from  AB  to  BE  in  the  same  manner,  as 
from  AB  to  BC  in  the  last  case,  then  remove  to  E, 
and  continue  the  process  by  deflecting  from  BE  to 
EC,  and  so  on,  until  the  instrument  is  brought  into 
the  position  CD,  the  index  will  then  show  the  angle 
as  before. 


Fig-.  7 


METHODS  OF  LOCATION,  &C.  13 


PROPOSITION  IV. 


To  run  a  Line  from  a  given  Point,  parallel  to  a 
given  line. 

LET  C,  Fig.  6th  and  7th,  be  the  point,  and  AB  the 
given  line,  set  the  instrument  at  B  and  proceed  as  di- 
rected in  the  second  and  third  cases  of  the  last  propo- 
sition until  arriving  at  C,  then  turn  the  index  back  to 
Zero  and  the  glass  will  then  be  in  the  line  CF  parallel 
to  AB. 


OF  THE 

(  UNIVERSITY   ) 

OF 


- 

METHODS  OP  LOCATION,  &C.  15 


PROPOSITION  V. 


To  describe  a  circle,  on  the  ground  with  the  Transit 
and  Chain  from  any  point  in  a  given  Tangent 
with  any  degree  of  curvature  or  central  angle. 

LET  HG,  Fig.  8,  be  the  tangent,  and  A  the  point. 
Place  the  instrument  over  A,  set  the  index  to  Zero, 
and  take  a  back  sight  to  H.  Then  reverse  the  Tele- 
scope and  turn  the  index  till  it  shows  the  tangential, 
i.  e.  half  the  central  angle,  and  measure  AB  with  the 
chain. 

Make  BAG,  CAD,  and  DAE,  successively,  equal  to 
the  tangential  or  circumferential  angle  and  measure 
BC,  CD,  and  DE  as  before  ;  B,  C,  D  and  E  will  be 
points  in  the  curve. 

When  E  is  found,  the  index  shows  the  whole  angle 
EAG  ;  then  remove  the  instrument  to  E,  take  a  back 
sight  to  A,  and  turn  the  index  till  it  shows  twice  the 
angle  EAG ;  then  turn  the  glass  toward;?  F,  and  E  F 
will  be  the  course  of  a  tangent  from  E.  This  is  called 
completing  the  tangent. 

If  it  be  necessary  to  continue  the  curve,  repeat  the 
above  process  by  adding  the  tangential  angle  succes- 
sively, and  measuring  EL,  LM,  &c.  as  before.  To 
complete  the  tangent  at  M,  after  taking  a  back  sight 
from  M  to  E,  it  is  necessary  to  add  only  the  tangen- 
tial angle  FEM  to  what  is  already  shown  by  the  index, 
which  will  then  show  the  whole  central  angle  ASM, 
or  exterior  angle  of  the  tangents  HA  and  MO.  But 
if  the  proper  course  of  the  tangents  require  it  to  be 


16  METHODS  OF  LOCATION 

drawn  from  a  point  between  E  and  L,  ascertain  by 
Proposition  III,  page  11,  the  angle  it  would  make  with 
EF,  and  make  VEF  equal  to  half  that  angle,  and 
take  EV  :  EL  :  :  VEF  :  LEF,  then  make  IVK 
equal  to  VEF  and  VK  is  the  course  of  the  tangent 
required. 

If  the  angle  VEF  were  known,  that  is,  if  the  whole 
exterior  angle  of  VK  with  HA  were  known,  the  point 
V  may  be  fixed  from  A,  by  making  VAE  equal  to 
VEF,  i.  e.  VAG  equal  to  half  the  exterior  angle  of  VK 
with  AG,  and  measuring  EV  as  before. 

Intermediate  points  a,  6,  c,  must  be  found  by  calcu- 
lating the  ordinates  by  Proposition  XV,  page  39,  or 
they  may  be  found  by  making  the  angles  proportionate 
to  the  distances  measured  on  AB,  and  setting  off  a.  b.  c. 
&c.  at  right  angles,  but  the  ordinates  are  best  suited  to 
an  unpractised  hand. 

The  demonstration  of  the  above  method  is  evident 
from  the  preliminary  propositions. 

NOTE.  Since  the  chain  EL  whose  central  angle  is  m  contains  100 
links,  each  link  of  EV  will  have  a  central  angle  of  .  m  . 

The  index  should,  therefore,  be  divided  into  hundredths  instead  of 
minutes.  More  perplexity  will  he  avoided  by  this  simple  contrivance 
than  can  well  be  imagined  by  any  one  who  has  not  made  the  experi- 
ment. 


A  G 


Fijr.10 


FOR  RAILWAY  ENGINEERS.  17 


PROPOSITION  VI. 

To  change,  the,  origin  of  a  Curve  so  that  it  shall  ter- 
minate in  a  tangent,  parallel  to  a  given  Tan- 
gent. 

SUPPOSE  the  curve  AE,  Fig.  9th,  terminating  in  the 
tangent  EF  to  have  been  described  as  directed  in  the 
preceding  Proposition,  and  that  the  nature  of  the 
ground  requires  that  it  should  terminate  in  IK  parallel 
toEF. 

Measure  the  distance  El  on  the  line  parallel  to  AG, 
and  make  AG  equal  to  El,  G  will  be  the  origin  of  a 
curve,  similar  to  AE,  that  will  terminate  in  IK  at  the 
point  I. 

The  parallelism  of  all  the  lines  drawn  from  A  and  E, 
with  similar  lines  from  G  and  I,  sufficiently  show  the 
correctness  of  this  method  without  a  formal  demon- 
stration. 


PROPOSITION  VII. 

Having  a  curve  AE,  Fig.  10,  terminating  in  a  tan- 
gent EF,  it  is  required  to  find  where  a  curve  of  a 
different  radius  originating  in  the  same  point 
would  terminate  in  a  tangent  parallel  to  EF. 

CONSTRUCTION. 

LET  L  be  the  centre  of  AE,  and  on  AL  produced, 
take  A  M  equal  to  the  radius  of  the  other  circle,  draw 

2* 


18  METHODS  OF  LOCATION 

ES  parallel  and  equal  to  LM,  and  from  S  with  the 
radius  SE  describe  the  differential  circle  El,  join  MS 
and  produce  it  till  it  cuts  El  in  I,  then  will  I  be  the 
terminating  point  required. 

For  ES  being  equal  and  parallel  to  LM,  MS  must 
be  equal  and  parallel  to  LE  ;  hence,  MI  is  equal  to 
MA,  and  the  angles  LMS  and  ESI  are  equal  lo  ALE, 
and  therefore,  the  tangent  EF  is  parallel  to  IK. 

Hence,  the  field  voperation  is  evident.  The  instru- 
ment must  be  set  over  E,  and  the  Telescope  brought 
into  the  position  ER  parallel  to  AG.  From  ER  as  a 
tangent  describe  the  differential  curve  El,  and  make  it 
equivalent  to  ALE,  and  I  will  be  the  point  sought. 

By  completing  the  parallelogram  SEPI,  it  will  be 
perceived  that  El  may  be  described  in  the  opposite 
direction,  by  starting  off  the  tangent  EF,  instead  of 
ER,  and  also  if  AI  were  given,  and  E  required,  IE 
might  be  described  either  from  IK  or  IN  the  parallel 
to  AG. 

The  degree  of  curvature  for  El  is  found  thus  : 
a  =  57.30  m  and  n  =  curvatures  of  AE  and  AI  ; 


then  —  =  Radius  of  AE  and   —  Radius  of  AI,  and 
m  n 

their  difference  =    am~an  and  a  divided  by  this 

mn 
mn 

result    =    ;  that  is,  multiply  the  curvatures  to- 
rn— n'  ^J 

gether,  and  divide  by  their  difference. 

NOTE.  When  the  product  of  the  two  curvatures  is  very  great  and 
their  difference  small,  a  differential  curvature  will  result  which  is  too 
large  to  be  used  with  the  chord  of  100  feet  without  occasioning  a 
sensible  error  in  the  result  the  proper  method  then  is,  to  take  smaller 
chords  of  10  or  20  feet ;  reducing  the  curvature  in  a  like  proportion. 

Thus,  if  m  be  5  and  n  be  6     mn    will  be  30°;  then  instead  of  a  chord 

m — n 
of  100  feet,  take  20  feet  and  a  central  angle  of  6°. 


Fig-.  11 


FOR  RAILWAY  ENGINEERS.  19 


PROPOSITION  VIII. 


Having  a  Curve,  HAE,  (Fig.  11,)  and  a  line  NK  at 
a  distance  from  it,  to  d?aw  another  Curve  of  a  dif- 
ferent radius,  ivhich  shall  touch  the  first  Curve, 
and  also  the  line  NK. 

CONSTRUCTION. 

LET  L  be  the  centre  of  HAE.  Draw  the  tangent 
EF  parallel  to  IK,  and  through  L  and  E  draw  LS,  and 
make  it  equal  to  the  radius  of  the  other  circle  :  from 
S  with  the  radius  SE  describe  the  differential  circle 
El,  cutting  IK  in  I:  join  IS  and  draw  LA  parallel  to 
it ;  complete  the  parallelogram  ISLM;  then  will  M  be 
the  centre  of  the  circle  required.  For  MS  is  equal  to 
SI  or  SE,  and  LE  to  LA,  and  hence,  MA  is  equal  to 
LS  or  MI,  and  a  circle  described  from  M  with  radius 
MA  will  touch  HAE  in  A,  and  because  MI  and  LS 
are  parallel,  and  EF  and  IK  are  parallel,  the  angle 
MIK  is  equal  to  LEF,  which  is  a  right  angle,  and 
therefore  the  circle  AI  touches  IK  in  I. 

APPLICATION. 

It  is  evident  from  the  mathematical  solution,  that  the 
curve  HAE  must  be  continued  until  its  tangent  be- 
comes parallel  to  IK,  then  from  EF  as  a  tangent  the 
differential  curve  El  must  be  traced  as  directed  in  the 
preceding  proposition  until  it  intersects  IK  in  I,  the 
central  angle  ESI  must  then  be  ascertained  from  the 


20  METHODS  OF  LOCATION 

index,  (after  completing  the  tangent  as  in  Proposi- 
tion If,)  and  the  curve  A  E  made  equivalent  to  it;  A 
will  then  be  the  P.  C.  C.  sought,  from  which  if  a  curve 
be  described  with  the  proper  curvature,  it  will  touch 
IK  in  the  point  I. 

It  is  also  evident,  that  if  AI  were  given,  and  AE 
required  that  the  centre  S  would  fall  upon  N,  and  that 
IE  must  be  described  from  the  tangent  IK  in  the  op- 
posite direction  till  it  intersects  EF. 

NOTE.  The  observations  in  the  note  attached  to  the  preceding  Pro- 
position  will  apply  to  this,  as  well  as  several  other  succeeding  Propo- 
sitions. 

It  often  happens,  that  the  continuation  of  HAE  to  the  tangent  EF 
is  rendered  difficult  or  impracticable  by  the  roughness  of  the  ground; 
recourse  must  then  be  had  to  the  succeeding  Proposition. 


Fig-.  12 


itr  13 


IT 


FOR  RAILWAY  ENGINEERS.  21 


PROPOSITION  IX. 


Having  located  a  compound  Curve  HA,  AC,  Fig. 
12 ,  terminating  in  a  Tangent  CD,  it  is  required 
to  change  the  point  of  compound  curvature  from 
AtoH  so  that  the,  Curve  will  terminate  in  a  Tan- 
gent EH  parallel  to  CD. 

LET  I  be  the  centre  of  HBA,  and  L  the  centre  of 
AC,  and  draw  ILA,  then  IL  is  the  difference  of  the 
radii  of  the  two  curves  :  make  CM  equal  and  parallel 
to  IL,  and  describe  the  differential  curve  CE,  cut- 
ting EF  in  E.  Draw  IB  parallel  to  ME,  then  will  B 
be  the  new  P.  C.  C.  sought.  For  if  BG  be  made  equal 
to  LA  or  LC,  IG-  will  be  equal  and  parallel  to  ME, 
and  consequently  GL  will  be  equal  and  parallel  to  CE, 
and  GE  will  be  equal  and  parallel  to  LC,  and  at  right 
angles  to  EF,  and  hence,  a  circle  described  from  G 
with  the  radius  GB  will  make  tangent  upon  EF  at 
the  point  E. 

APPLICATION. 

If,in  starting  from  the  point  A,  the  index  of  the  Tran- 
sit be  set  at  Zero,  it  will  show  upon  completing  the 
tangent  at  C  the  whole  angle  between  AN  and  CD,  and 
by  turning  it  back  again  to  Zero,  the  Telescope  will 
assume  a  position  CO  parallel  to  AN:  then  from  CO 
as  a  tangent  describe  the  differential  curve  CE,  until  it 
intersects  EF,  and  ascertain  as  in  the  last  Proposition, 
the  central  angle  of  EC  which  is  equivalent  to  AB, 


22  METHODS  OF  LOCATION 

and  consequently  the  point  B  may  be  readily  obtained 
by  setting  the  transit  over  A,  and  retracing  a  portion 
of  the  curve  equivalent  to  EC. 

A  second  case  of  this  Proposition  occurs,  when  the 
radius  of  AC,  Fig.  13,  is  greater  than  that  of  HA,  the 
effect  of  which  is  to  throw  the  point  M  on  the  opposite 
side  of  EC,  and  consequently  the  curve  must  be  turned 
in  the  opposite  direction  in  passing  from  C  to  E. 


Fi§M4 


II 


FOR  RAILWAY  ENGINEERS.  23 


PROPOSITION  X. 

When  the  position  of  the  tangent  EF,  Fig.  14,  is 
such  that  if  produced  it  would  cut  HAB,  it  is 
evident  that  a  curve  which  shall  touch  them  both 
externally,  must  be  turned  in  the  opposite  direc- 
tion from  HAB.  This  is  termed  a  Reversed  Curve, 
and  may  be  described  as  follows. 

TAKE  a  point  H  in  HAB,  where  the  tangent  HP  is 
parallel  to  EF,  through  H  and  the  centre  L  draw 
HLS  and  make  it  equal  to  the  sum  of  the  two  radii 
HL  and  MI.  From  S,  with  radius  HS,  describe  the 
integral  curve  HE  intersecting  EF  in  E,  and  com- 
plete the  parallelogram  LSEM.  Then  since  ML 
equals  ES,  or  HS  and  LA  equals  HL,MA  must  equal 
ME  or  LS,  and  since  HP  is  parallel  to  EF,  and  ME 
to  HL,  the  angle  MEF  equals  LHP,  which  is  a  right 
angle.  Therefore  a  circle  described  from  the  centre 
M  with  radius  MA  will  touch  EF  in  E.  * 

The  field  operation  is  evident  from  the  above  con- 
struction, being  strictly  analogous  to  the  preceding 
Proposition.  The  integral  curvature  is  found  as  fol- 
lows :  a  =  57.30,  m  =  curvature  AE,  n  =  curva- 
ture HA,  then  radius  AE,  =  — ,  and  radius  HA  — 

m  n 

and  their  sum  = _ ,  and  a  divided  by  this  re- 

mn 

suit  gives  — mn  ;  i.  e.   multiply  the   curvatures  to- 
rn +  n 

gether  and  divide  by  their  sum. 


24  METHODS  OF  LOCATION 

When  H  is  inaccessible,  as  it  often  will  be,  the  Pro- 
position must  be  solved  by  the  method  immediately 
following. 


Fig-.  15 


11 


FOR  RAILWAY  ENGINEERS. 


SECOND  METHOD. 


TAKING  any  point  A,  Fig.  15th,  describe  AC  until 
the  tangent  becomes  parallel  to  EF,  then  from  0  and 
the  tangent  CO  parallel  to  AN  describe  the  integral 
curve  CE,  intersecting  EF  in  E ;  then  ascertain  the 
whole  central  angle  of  CE,  and  extend  HA  to  B,  so 
that  AB  shall  be  equivalent  to  CE,  and  B  will  be  the 
point  of  reversed  curvature  required. 

The  demonstration  of  this  method  is  so  analogous 
to  that  of  Proposition  VIII,  that  it  is  unnecessary  to  go 
into  it  here ;  it  depends  on  the  parallelism  of  the  lines 
connecting  different  parts  of  the  figure. 


or  THE         \ 
UNfVERSITY   I 

OF  .        / 

W  CAI    r-    .-,  "»h  Jr* 


Fig-.  17 


OF  THE 

UNIVERSITY 


METHODS  OF  LOCATION,  &C.  27 


PROPOSITION  XI. 


Having  located  a  Curve  HAB,  Fig.  16,  17  and  18, 
to  draw  through  a  point  C,  another  Curve  which 
shall  touch  HAB. 

IN  HAB  take  any  point  A,  and  through  A  and  the 
centre  I  draw  AL,  and  make  it  equal  to  the  radius  of 
the  required  circle,  and  from  L  as  centre,  describe 
AD  ;  join  1C  and  complete  the  parallelogram  1LMC, 
then  will  M  be  the  centre  of  an  integral  circle,  Fig.  16, 
and  a  differential  circle,  Fig.  17  and  18.  From  the 
centre  M  describe  CD  cutting  AD  in  D,  join  MD  and 
draw  IB  parallel  to  it,  make  BG  equal  to  LA  then 
will  G  be  the  centre  of  the  curve  BC. 

For,  since  IL  and  IG  are  parallel  and  equal  to  MC 
and  MD,  GS  must  be  parallel  and  equal  to  CD,  and 
hence,  GC  must  be  parallel  and  equal  to  LD,  i.  e.  to 
LA  or  GB  ;  hence,  a  circle  described  from  G  with 
GB  will  pass  through  C  as  required. 

APPLICATION. 

Since  CM  is  parallel  to  AI,  the  tangent  CN  must 
be  parallel  to  AR.  Therefore,  from  any  point  A  de- 
scribe AD  with  the  Transit,  and  from  any  conve- 
nient point  in  AD  deflect  to  C,  and  by  Proposition  IV, 
page  13,  bring  the  Telescope  into  the  position  CN,  pa- 
rallel to  AR,  then  describe  the  differential  or  integral 
curve  CD  till  it  intersects  AD  ;  complete  the  tangent 
AD,  and  ascertain  the  central  angle  of  CD  and  make 


28  METHODS  OF  LOCATION 

AB  equivalent :  from  a  tangent  at  B  describe  BC, 
which  must  pass  through  the  point  C. 

Observe  that  in  Fig.  17,  CD  is  turned  towards  the 
same  hand  as  AB,  and  in  Fig.  16  and  18  towards  the 
opposite  hand.  The  reason  is  sufficiently  evident  from 
the  position  of  ttye  parallelogram  ILMC. 

The  distinction  of  the  several  cases  must  be  care- 
fully observed.  In  Fig.  16  ABC  is  a  reversed  curve. 
In  Fig.  17  and  18  ABC  is  a  compound  curve.  In 
Fig.  17  C  is  outside  of  AB  and  the  radius  of  BC  is  of 
course  the  greatest.  In  Fig.  18  C  is  inside  of  AB,  and 
of  course  the  radiaus  of  BC  must  be  least. 


Fig-.  19 


FOR  RAILWAY  ENGINEERS.  29 


PROPOSITION  XII. 


To  draw  a   Tangent  to  a  Curve  from  a  Point 
without  it. 

LET  HA,  Fig.  19,  be  the  curve  and  C  the  point  with-* 
out  it,  and  let  S  be  the  centre  of  HA.     Join  CS  and 
on  it  describe  the  semicircle  CAS,  then  will  CS  be 
the  tangent  required  ;  for  the  angle  CAS,  being  in  a 
semicircle,  must  be  a  right  angle. 

APPLICATION. 

The  centres  of  Railway  Curves  are  always  too  re- 
mote to  be  useful  for  any  purpose  of  location,  and 
therefore,  other  methods  must  be  resorted  to  in  all 
cases,  where  the  position  of  their  centres  with  regard 
to  objects  beyond  their  circumferences  are  involved. 

In  the  present  case  take  any  point  H  in  the  curve 
and  measure  CH  ;  from  H  towards  the 'centre  lay  off 
HS,  any  convenient  multiple  of  the  radius,  and  make 
He  a  similar  multiple  of  HC,  then  will  cs  be  a  similar 
multiple  of  CS,  and  by  the  similarity  of  triangles  is 
evidently  parallel  to  it.  Divide  57.30  by  \  of  CS,  and 
the  quotient  will  be  the  curvature  of  CAS.  With  this 
curvature  and  from  a  tangent  at  right  angles  to  Cs  or 
CS,  describe  CAS,  and  where  it  cuts  HA  will  be  the 
P.T.  required. 

3* 


METHODS  OF  LOCATION,  &C.  31 


PROPOSITION  XIII. 


Between  two  Curves  already  described,  to  draw  a 
third  Curve  with  a  given  Radius  which  shall 
touch  them  both. 


CASE  FIRST. 


When  all  the  Curves  must  be  turned  in  the  same 
direction. 

LET  HS  and  LM,  Fig.  20,  be  the  given  curves  of 
which  A  and  B  are  the  centres,  it  is  required  to  de- 
scribe a  third  curve  SFM,  with  a  given  radius  which 
shall  touch  the  first  two. 

Take  any  point  H  in  HS,  and  a  corresponding  point 
L  in  LM  ;  draw  AH  and  BL,  and  produce  them  till 
AC  and  BD  are  each  equal  to  the  radius  of  the  third 
curve.  From  C  and  D  as  centres  describe  the  diffe- 
rential curves  HF  and  LF,  and  from  their  intersection 
draw  FE  parallel  to  AC  and  BD,  and  complete  the 
parallelogram  ACFE  or  BDFE,  then  will  E  be  the 
centre  of  a  curve  SFM  which  shall  touch  the  other 
curves  in  S  and  M. 

For  AE  being  equal  to  CF  or  CH,and  AS  to  AH,  ES 
must  be  equal  to  AC,  which  by  construction  is  equal 
to  EF.  Therefore  EF  is  equal  to  ES,  and  by  the 


32  METHODS  OF  LOCATION 

same  reasoning  EM  may  be  proved  to  be  equal  to  EF; 
therefore  E  is  the  centre  of  the  curve  SFM,  and  since 
SE  and  EM  pass  through  the  centres  A  and  B,  SFM 
must  touch  HS  and  LM  in  S  and  M  respectively. 

APPLICATION. 

Since  FE  is  parallel  to  HA,  the  tangent  FN  must 
be  parallel  to  HR.  Therefore  from  any  two  corres- 
ponding points,H  and  L,describe  the  differential  curves 
HF  and  LF  and  from  their  intersection  with  a  tangent 
parallel  to  HR  describe  the  required  curve  each  way 
till  it  touches  the  others  as  required. 


D 


FOR  RAILWAY  ENGINEERS.  33 


CASE  SECOND. 


When  HS  and  NIL,  Fig.  21,  are  turned  in  the  same 
direction  and  it  is  required  that  SM  shall  be 
turned  in  the  contrary  direction. 

JOIN  HA  and  LB  as  before,  and  produce  them  till 
AC  and  BD  are  equal  to  the  radius  of  the  third  circle. 

Describe  the  integral  circles  HF,  LF  ;  draw  FE 
and  complete  the  parallelograms  as  before,  and  E  will 
be  the  centre  of  the  circle  required. 

The  demonstration  of  this  case  is  so  analogous  to 
the  last,  that  it  need  not  be  inserted  here. 

The  field  of  operation  is  also  analogous  to  the  last. 
The  integral  curves  HF  and  LF  must  be  turned  in  the 
same  direction  as  HS  or  LM. 


OF  THE 

UNIVERSITY 

OF 


Fig.  22 


METHODS  OF  LOCATION,  &C.  35 


CASE  THIRD. 


Where  HS  and  ML,  Fig.  22,  are  turned  in  opposite 
directions. 

TAKE  H  and  L  as  before,  and  draw  CA  and  BD  in 
such  a  manner  that  CH  shall  be  the  radius  of  an  inte- 
gral and  DL  of  a  differential  curve,  to  HS  and  LM 
respectively,  complete  the  parallelograms  as  before, 
and  E  will  be  the  centre  of  the  circle  required,  which 
must  be  turned  in  the  same  direction  as  the  curve 
to  which  the  differential  radius  was  applied. 

NOTE.  It  will  be  observed,  generally,  of  these  three  cases,  that  in- 
tegral curves  are  turned  in  the  same  direction  as  the  primary  curves, 
and  differential  curves  in  the  opposite  direction,  and  that  the  curve 
SM  is  always  turned  in  the  opposite  direction  to  the  auxiliary  curves 
used  in  constructing  it. 


Fig-.  24 


METHODS  OF  LOCATION,  &C.  37 


PROPOSITION  XIV. 


To  draw  a  Tangent  to  two  Curves  already  located. 

TAKE  any  point  H,  (Fig.  23  and  24,)  in  the  circle 
HS,  and  take  L  a  corresponding  point  in  the  other 
circle  LN,  and  measure  HL.  Then  suppose  HI 
drawn  parallel  to  AB  and  take  CH  :  HL  :  :  HA  :  LI, 

That  is  in  Fig.  23,  CH  =  HL  x  >— — 


m — n 
n 


And  in  Fig.  24,  CH  =  HL  x 

m+n 

Then  C  will  be  a  point  in  the  common  Tangent 
which  may  be  drawn  as  directed  in  Proposition  XII. 

For  HAC  and  LIH  are  similar  triangles,  and  LI  : 
HA  :  :  HI  :  CA,  but  LI,  HA  and  HI  are  constant 
and,  therefore,  C  is  constant,  and  any  line  drawn 
through  C  to  meet  both  circles  will  make  the  same 
angle  with  the  radii  at  the  point  of  meeting,  and  there- 
fore, if  it  touch  one  circle,  it  will  touch  both. . 

NOTE.     If  HA  =  LB,then  CSN  (Fig.  23,)  will  be  parallel  to  HL. 


Fig-.  25 


METHODS  OP  LOCATION,  &C.  39 


PROPOSITION  XV. 


To  find  the  Ordinal le  EF,  Fig.  25,  at  any  point  in 
a  given  chord  KB,  the  diameter  of  the  circle  being 
known. 


114.6 

Now  if  ab  be  given  in  parts  of  a  chain  of  66  feet 
and  the  value  of  x  be  required  in  inches,  then  x  = 

792  * 

abm  x  =  cibm  X  6.9. 

114.6 

But  if  the  chain  be  100  feet  and  x  be  required  as 

before,  then  x  =  abm  x  =  abm  X   10.5. 

114.6 

And  if  the  chain  be  100  feet,  and  x  be  required  in 

feet,  then  x  =  abm  x  =  abm  X  .875. 

114.6 

NOTE.  It  must  be  remembered  (see  Explanations,  No.  2,)  that  the 
chord  of  100  parts  is  considered  as  a  unit,  and  therefore,  the  decimal 
point  in  a  and  b  must  be  so  placed  as  to  diminish  their  value  in  like 
proportion. 


PUT  AE  =  a,  EB  =  b,  GH  =  v,  EF  =  x  and 
D  =  diameter.     Then,  EC  =  D — 2  v — x  and  ab  =  Dx 

— 2vx — a?". 

Now  2  vx — a?  is  too  small  to  affect  the  result  in  any 
case  of  location  ;  it  may  therefore  be  omitted  and  the 

equation  becomes  ab  =  Da?.     But  D  = — ;  there- 

m 

fore  ab  = —-,  and  x  ==  ab  x   — =  abm  x 

m  114.6 


40  METHODS  OF  LOCATION 


PROPOSITION  XVI. 


To  measure  the  Width  of  a  River  or  distance  to  any 
inaccessible  object  in  the  line  of  survey  . 

i 

LET  AB,  Fig.  26,  be  part  of  a  line  of  survey  in 
which  A  and  B  are  on  opposite  sides  of  a  river. 

From  A  at  right  angles  to  AB  lay  off  any  conve- 
nient distance  AC,  so  that  B  may  be  seen  from  C  ; 
remove  the  instrument  to  C,  and  lay  off  CD  at  right 
angles  to  CB,  fixing  the  point  D  in  AB  produced,  and 
measure  DA. 

Then  AB  is  a  third  proportional  to  DA  and  AC  and 

\C3 
of  course  J_==AB. 


L  _ 


On  the  opposite  page  will  be  found  the  table  of 
chords  referred  to  in  the  preface.  The  numbers  in 
the  table  are  the  ratios  of  the  base  to  the  side  of  an 
isosceles  triangle  for  every  degree  of  vertical  angle. 
It  would  be  difficult  to  give  a  specific  account  of  its 
various  uses,  since  it  may  be  applied  to  every  case 
which  can  be  resolved  into  an  isosceles  triangle  of 
which  one  side  and  angle  are  known. 


FOR  RAILWAY  ENGINEERS. 


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